Torus initial PR

properties/P000122.md

@@ -21,3 +21,9 @@ $\mathbb R^0=\{0\}$ is an isolated point.
 
 The closely related {P123} property is
 defined on page 316 of {{zb:0951.54001}} and on page 38 of {{mr:2766102}}.
+
+----
+#### Meta-properties
+
+- This property is preserved by finite products.
+- This property is preserved by arbitrary disjoint unions.

properties/P000184.md

@@ -16,3 +16,4 @@ Per Theorems 1.1.2 and 1.11.4 of {{zb:0401.54029}}, this is equivalent to being
 #### Meta-properties
 
 - This property is hereditary.
+- This property is preserved by finite products.

spaces/S000224/README.md

@@ -0,0 +1,14 @@
+---
+uid: S000224
+name: Torus
+refs:
+  - wikipedia: Torus
+    name: Torus on Wikipedia
+---
+
+The product space {S170} $\times$ {S170}.
+
+Equivalently, the quotient of
+$[0,1]^2$ obtained by identifying $(x,0)\sim(x,1)$ and $(0,y)\sim(1,y)$.
+
+Homer Simpson's favourite food.

spaces/S000224/properties/P000016.md

@@ -0,0 +1,7 @@
+---
+space: S000224
+property: P000016
+value: true
+---
+
+$X$ is the square of {S170} and {S170|P16}.

spaces/S000224/properties/P000037.md

@@ -0,0 +1,7 @@
+---
+space: S000224
+property: P000037
+value: true
+---
+
+$X$ is the square of {S170} and {S170|P37}.

spaces/S000224/properties/P000065.md

@@ -0,0 +1,7 @@
+---
+space: S000224
+property: P000065
+value: true
+---
+
+Immediate from the definitions.

spaces/S000224/properties/P000089.md

@@ -0,0 +1,7 @@
+---
+space: S000224
+property: P000089
+value: false
+---
+
+The the function $(z,w)\mapsto(-z,w)$ is continuous self-map with no fixed points.

spaces/S000224/properties/P000122.md

@@ -0,0 +1,7 @@
+---
+space: S000224
+property: P000122
+value: true
+---
+
+$X$ is the square of {S170} and {S170|P122}.

spaces/S000224/properties/P000155.md

@@ -0,0 +1,7 @@
+---
+space: S000224
+property: P000155
+value: false
+---
+
+Since {S170|P155}, $X$ is locally homeomorph to {S176}, which is not homeomorph to {S25}.

spaces/S000224/properties/P000184.md

@@ -0,0 +1,10 @@
+---
+space: S000224
+property: P000184
+value: true
+refs:
+  - wikipedia: Torus
+    name: Torus on Wikipedia
+---
+
+$X$ is the square of {S170} and {S170|P16}.

spaces/S000224/properties/P000200.md

@@ -0,0 +1,10 @@
+---
+space: S000224
+property: P000200
+value: false
+refs:
+  - zb: "1044.55001"
+    name: Algebraic Topology (Hatcher)
+---
+
+We have $\pi_1(X)\cong\pi_1(S^1\times S^1) \equiv \pi_1(S^1)\times \pi_1(S^1)\cong \mathbb{Z}\times\mathbb{Z}$.